It is known that the generating function f of a sequence of Toeplitz matrices {T_n(f)}_n may not describe the asymptotic distribution of the eigenvalues of T_n(f) if f is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of T_n(f) are real for all n, then they admit an asymptotic expansion of the same type as considered in previous works, where the first function, called the eigenvalue symbol \(\mathfrak {f}\), appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of T_n(f). This eigenvalue symbol \(\mathfrak {f}\) is in general not known in closed form. After validating this working hypothesis through a number of numerical experiments, we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function \(\mathfrak {f}\). The proposed algorithm, which opposed to previous versions, does not need any information about neither f nor \(\mathfrak {f}\) is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of \(\mathfrak {f}\). Future research directions are outlined at the end of the paper.

众所周知，如果f不是实数，则一系列 Toeplitz 矩阵 { T _n ( f )} _n的生成函数f可能无法描述T _n ( f )的特征值的渐近分布。在本文中，我们假设一个工作假设是，如果T _n ( f )的特征值对于所有n都是实数，那么它们承认与之前工作中考虑的相同类型的渐近展开，其中第一个函数称为特征值符号\(\mathfrak {f}\)，出现在这个展开式中是实数，它描述了T _n ( f )的特征值的渐近分布。这个特征值符号\(\mathfrak {f}\)通常在封闭形式中是未知的。在通过大量数值实验验证了这个工作假设之后，我们提出了一种无矩阵算法来近似特征值分布函数\(\mathfrak {f}\)。与以前的版本相反，所提出的算法不需要关于f和\(\mathfrak {f}\) 的任何信息，并在广泛的数值示例上进行测试；在某些情况下，我们甚至可以找到\(\mathfrak {f}\)的解析表达式. 论文末尾概述了未来的研究方向。